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Dictionary of the History of Ideas | ||

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*II*

At the start of the modern period, the instrumental

and exemplary nature of
mathematics recognized by

the new science led to extending the
mathematical

mode of exposition to various disciplines. This occurred

first in the extension of the work begun by the Greeks

to the science of
nature and, more exactly, to that part

which is generally regarded as its
foundation, namely,

Mechanics. Galileo was inspired by the method of

Archimedes, and tried to do for Dynamics what Archi-

medes had done for Statics. Descartes, in his *Principles of Philosophy* (

*Principia philosophiae,*1644), postulated

three “laws of nature” dealing with motion, justifying

them

*a priori*through God's perfection, and claiming

that he could demonstrate all of physics by means of

these three laws. Finally, and above all, Newton in his

*Mathematical Principles of Natural Philosophy*(

*Philo-*

sophiae naturalis principia mathematica,1687), orga-

sophiae naturalis principia mathematica,

nized Mechanics in the form of a logical system which

has remained classical. It was taught often best, espe-

cially in France, as a mathematical discipline. Newton's

work opens with the statement of eight definitions and

three axioms or laws of motion, starting from which

However, the prestige of the Euclidean axiomatic

model was such that after
going beyond mathematics,

it won over disciplines which are outside of
science

properly speaking. Descartes, while maintaining his

preference
for the analytical order of his *Meditations,*

had
already agreed, to satisfy the authors of the *Second
Objections,* to expound in synthetic order the “reasons

which prove the existence of God and the distinction

between the mind and the human body, the reasons

arranged in a geometric manner,” demonstrating his

propositions through definitions, postulates, and

axioms. His example was followed by Spinoza, with

a breadth and rigor which fascinated many minds, in

his

*Ethics, demonstrated in a geometric order*(

*Ethica*

ordine geometrica demonstrata,1677); Spinoza's work

ordine geometrica demonstrata,

was expounded by subjecting it, from one end to the

other with no exceptions, to the requirements of Eu-

clidean standards with definitions, postulates, and

axioms followed by propositions, demonstrations,

corollaries, lemmas, and scholia.

Jurisprudence, along with metaphysics and ethics,

also entered upon the road
of axiomatization. When-

ever Leibniz wished to
give examples of disciplines

containing rigorous reasoning he mentioned the
works

of the Roman jurisconsults as well as of the Greek

mathematicians. He offered an example himself of a

juridical exposition by
definitions and theorems in his

sample of legal persuasion or demonstration
(*Specimen certitudinis seu demonstrationum in
jure,* 1669) in which

he refers to “those ancients who arranged their rebut-

tals by means of very certain and quasi-mathematical

demonstrations.” Not long before, Samuel von Pufen-

dorf had published his

*Elementa jurisprudentia uni-*

versalis(1660), written under the double inspiration

versalis

of Grotius and his own teacher Weigel who taught both

law and mathematics. Pufendorf wished to show that

law, rising above historical contingencies, contains a

body of propositions which are perfectly certain and

universally valid, and capable of being made the con-

clusions of a demonstrative science. As a matter of fact,

here, as in Leibniz, axiomatization was still only

making a start. Instead of producing the propositions

and their proofs as logical consequences of principles,

Pufendorf presented them substantially in extensive

commentaries which follow each one of his twenty-one

definitions in order to avoid, he said, “a certain aridity

which might have run the risk of distorting this disci-

pline if we had presented it by cutting it up into small

parts, as is the manner of mathematics.” In the wake

of Pufendorf the so-called school of “natural law and

human right” elaborated for more than a century theo-

ries in which “one deduces through a continuous chain

leading from the very nature of man to all his obliga

tions and all his rights,” restating the subtitle of one

of Christian Wolff's works. Wolff, as a disciple of

Leibniz, boasted of accomplishing what others had only

proposed to do, namely, to deal with the theory of

human actions according to the demonstrative method

of the mathematicians (

*Philosophia practica universalis,*

methodo scientifica pertracta,Frankfurt and Leipzig,

methodo scientifica pertracta,

1738-39). Nevertheless, here also, we are quite far from

the logical rigor and even the mode of presentation

of Euclidean geometry.

Dictionary of the History of Ideas | ||